Question

If $ 3x+5y=9 $ and $ 5x+3y=7 $ then what is the value of $ x+y? $

Answer 1

Hint: To solve the question given above, we will first solve the given pair of linear equations in two variables with the help of method of substitution. With the help of this, we will find the values of x and y and then we will put these values of x and y into the term $ \left( x+y \right) $ to get an answer.

Complete step-by-step answer:
In the question, we are given a pair of linear equations in two variables. Before solving the question, we must know what is a linear equation in two variables. Linear equation in two variables is a type of equation which contains the terms of x and y having single power joined by addition or subtraction. The general form of linear equation in two variables is $ ax+by+c=0 $ . Now, it is given in the question that, we have to find the values of $ \left( x+y \right) $ . For this, we will have to find the value of x and y separately. For this, we will use a substitution method. The equations given in question are:
  $ 3x+5y=9......\left( 1 \right) $
  $ 5x+3y=7.....\left( 2 \right) $
From equation (1), we can say that:
  $ 3x+5y=9 $
  $ \Rightarrow 3x=9-5y $
  $ \Rightarrow x=\dfrac{9-5y}{3}.......\left( 3 \right) $
Now, we will put the value of x form equation (3) into equation (2). After doing this, we will get:
  $ \Rightarrow 5\left( \dfrac{9-5y}{3} \right)+3y=7 $
  $ \Rightarrow \dfrac{45-25y}{3}+3y=7 $
  $ \Rightarrow 45-25y+9y=21 $
  $ \Rightarrow 45-21=25y-9y $
  $ \Rightarrow 24=16y $
  $ \Rightarrow y=\dfrac{24}{16} $
  $ \Rightarrow y=\dfrac{3}{2} $
Now, we will put this value of y from (4) to(3). After doing this, we will get:
  $ \Rightarrow x=\dfrac{9-5\left( \dfrac{3}{2} \right)}{3} $
  $ \Rightarrow x=\dfrac{9-\dfrac{15}{2}}{3} $
  $ \Rightarrow x=\dfrac{\dfrac{18-15}{2}}{3} $
  $ \Rightarrow x=\dfrac{\left( \dfrac{3}{2} \right)}{3} $
  $ \Rightarrow x=\dfrac{1}{2} $
The value of $ \left( x+y \right)=\dfrac{3}{2}+\dfrac{1}{2} $
  $ \Rightarrow \left( x+y \right)=\dfrac{4}{2} $
  $ \Rightarrow x+y=2 $

Note: The pair of linear equations given in question can also be solved with the help of graphical methods. In this method, we will plot the graph of these lines and find the intersection point of these two lines. The intersection point will be the solution of these equations.